Skip to main content

Advertisement

SpringerLink
Log in

A coupling extended multiscale finite element and peridynamic method for modeling of crack propagation in solids

  • Original Paper
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

A coupling extended multiscale finite element and peridynamic method is developed for the quasi-static mechanical analysis of large-scale structures with crack propagation. Firstly, a novel incremental peridynamic (PD) formulation based on the ordinary state-based PD model is derived utilizing the Taylor expansion technique. To combine the high computational efficiency of the EMsFEM and advantages of dealing with discontinuous problems of the PD, a coupling strategy based on the numerical base function is proposed, in which the displacement constraint relationships between the coarse element nodes of the EMsFEM and the material points of the PD among the coupling domain are constructed by the numerical base functions and are represented by a coupling strain energy function using the Lagrange multiplier method. Then, a bilinear softening material model is adopted to describe the damage and failure of the bond, and the incremental-iterative algorithms are applied to obtain the steady-state solutions. Finally, several representative numerical examples are presented, and the results demonstrate the accuracy and efficiency of the proposed coupling method for the quasi-static mechanical analysis of large-scale structures with crack propagation. Comparing with the single EMsFEM and PD method, the present coupling method can reduce much computational cost and well deal with crack problems, simultaneously.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26

Similar content being viewed by others

References

  1. Inglis, C.E.: Stresses in a plate due to the presence of cracks and sharp corners. Trans. Inst. Nav. Arch. 55, 219–241 (1913)

    Google Scholar 

  2. Griffith, A.A.: The phenomena of rupture and flow in solids. Philos. Trans. R. Soc. Lond. 211, 163–198 (1920)

    Google Scholar 

  3. Irwin, G.R.: Fracture Dynamics. In: Fracturing of Metals. American Society of Metals, Cleveland (1948)

    Google Scholar 

  4. Orowan, E.: Fracture Strength of Solids. In: Report on Progress in Physics. Phys. Soc. Lond. 12, 185–232 (1949)

    Google Scholar 

  5. Xie, D., Waas, A.M.: Discrete cohesive zone model for mixed-mode fracture using finite element analysis. Eng. Fract. Mech. 73(13), 1783–1796 (2006)

    Google Scholar 

  6. Schrefler, B.A., Secchi, S., Simoni, L.: On adaptive refinement techniques in multi-field problems including cohesive fracture. Comput. Methods Appl. Mech. Eng. 195(4), 444–461 (2006)

    MATH  Google Scholar 

  7. Melenk, J.M., Babuska, I.: The partition of unity finite element method: basic theory and applications. Comput. Methods Appl. Mech. Eng. 139, 289–314 (1996)

    MathSciNet  MATH  Google Scholar 

  8. Belytschko, T., Black, T.: Elastic crack growth in finite elements with minimal remeshing. Int. J. Numer. Methods Eng. 45(5), 601–620 (1999)

    MATH  Google Scholar 

  9. Dolbow, J., Moës, T., Belytschko, T.: Discontinuous enrichment in finite elements with a partition of unity method. Finite Elem. Anal. Des. 36(3–4), 235–260 (2000)

    MATH  Google Scholar 

  10. Strouboulis, T., Babuska, I., Copps, K.: The design and analysis of the generalized finite element method. Comput. Methods Appl. Mech. Eng. 181, 43–69 (2000)

    MathSciNet  MATH  Google Scholar 

  11. Strouboulis, T., Copps, K., Babuska, I.: The generalized finite element method. Comput. Methods Appl. Mech. Eng. 190, 4081–4193 (2001)

    MathSciNet  MATH  Google Scholar 

  12. Simo, J.C., Rifai, M.S.: A class of mixed assumed strain methods and the method of incompatible modes. Int. J. Numer. Methods Eng. 29, 1595–1638 (1990)

    MathSciNet  MATH  Google Scholar 

  13. Oliver, J., Huespe, A.E., Samaniego, E.: A study on finite elements for capturing strong discontinuities. Int. J. Numer. Methods Eng. 56, 2135–2161 (2003)

    MathSciNet  MATH  Google Scholar 

  14. Foster, C.D., Borja, R.I., Regueiro, R.A.: Embedded strong discontinuity finite elements for fractured geomaterials with variable friction. Int. J. Numer. Methods Eng. 72, 549–581 (2007)

    MathSciNet  MATH  Google Scholar 

  15. Linder, C., Armero, F.: Finite elements with embedded strong discontinuities for the modeling of failure in solids. Int. J. Numer. Methods Eng. 72, 1391–1433 (2007)

    MathSciNet  MATH  Google Scholar 

  16. Jiasek, M.: Comparative study on finite elements with embedded discontinuities. Comput. Methods Appl. Mech. Eng. 188, 307–330 (2000)

    Google Scholar 

  17. Lu, M.K., Zhang, H.W., Zheng, Y.G., Zhang, L.: A multiscale finite element method with embedded strong discontinuity model for the simulation of cohesive cracks in solids. Comput. Methods Appl. Mech. Eng. 311, 576–598 (2016)

    MathSciNet  Google Scholar 

  18. Miehe, C., Hofacker, M., Welschinger, F.: A phase field model for rate-independent crack propagation: robust algorithmic implementation based on operator splits. Comput. Methods Appl. Mech. Eng. 199, 2765–2778 (2010)

    MathSciNet  MATH  Google Scholar 

  19. Miehe, C., Welschinger, F., Hofacker, M.: Thermodynamically consistent phase-field models of fracture: variational principles and multi-field FE implementations. Int. J. Numer. Methods Eng. 83(10), 1273–1311 (2010)

    MathSciNet  MATH  Google Scholar 

  20. Verhoosel, C., Borst, R.: A phase-field model for cohesive fracture. Int. J. Numer. Methods Eng. 96, 43–62 (2013)

    MathSciNet  MATH  Google Scholar 

  21. Silling, S.A.: Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids 48, 175–209 (2000)

    MathSciNet  MATH  Google Scholar 

  22. Silling, S.A., Askari, E.: A meshfree method based on the peridynamic model of solid mechanics. Comput. Struct. 83(17–18), 1526–1535 (2005)

    Google Scholar 

  23. Silling, S.A.: Linearized theory of peridynamic states. J. Elast. 99, 85–111 (2010)

    MathSciNet  MATH  Google Scholar 

  24. Silling, S.A., Lehoucq, R.B.: Peridynamic theory of solid mechanics. Adv. Appl. Mech. 44, 73–168 (2010)

    Google Scholar 

  25. Silling, S.A., Epton, M., Wechner, O., Xu, J., Askari, E.: Peridynamic states and constitutive modeling. J. Elast. 88, 151–184 (2007)

    MathSciNet  MATH  Google Scholar 

  26. Ren, H.L., Zhuang, X.Y., Cai, Y.C., Rabczuk, T.: Dual-horizon peridynamics. Int. J. Numer. Methods Eng. 108, 1451–76 (2016)

    MathSciNet  Google Scholar 

  27. Le, Q.V., Chan, W.K., Schwartz, J.: A two-dimensional ordinary, state-based peridynamic model for linearly elastic solids. Int. J. Numer. Methods Eng. 98, 547–561 (2014)

    MathSciNet  MATH  Google Scholar 

  28. Ren, H.L., Zhuang, X.Y., Rabczuk, T.: Dual-horizon peridynamics: a stable solution to varying horizons. Comput. Methods Appl. Mech. Eng. 318, 762–782 (2017)

    MathSciNet  Google Scholar 

  29. Rabczuk, T., Zi, G., Bordas, S., Nguyen-Xuan, H.: A simple and robust three-dimensional cracking-particle method without enrichment. Comput. Methods Appl. Mech. Eng. 199, 2437–2455 (2010)

    MATH  Google Scholar 

  30. Rabczuk, T., Belytschko, T.: Cracking particles: a simplified meshfree method for arbitrary evolving cracks. Int. J. Numer. Methods Eng. 61, 2316–2343 (2004)

    MATH  Google Scholar 

  31. Areias, P., Msekh, M.A., Rabczuk, T.: Damage and fracture algorithm using the screened Poisson equation and local remeshing. Eng. Fract. Mech. 158, 116–143 (2016)

    Google Scholar 

  32. Areias, P., Reinoso, J., Camanho, P.P., Cesar de Sa, J., Rabczuk, T.: Effective 2D and 3D crack propagation with local mesh refinement and the screened Poisson equation. Eng. Fract. Mech. 189, 339–360 (2018)

    Google Scholar 

  33. Bobaru, F., Duangpanya, M.: A peridynamic formulation for transient heat conduction in bodies with evolving discontinuities. J. Comput. Phys. 231, 2764–2785 (2012)

    MathSciNet  MATH  Google Scholar 

  34. Oterkus, S., Madenci, E.: Peridynamic modeling of fuel pellet cracking. Eng. Fract. Mech. 176, 23–37 (2017)

    Google Scholar 

  35. Lai, X., Ren, B., Fan, H.F., Li, S.F., Wu, C.T., Regueiro, R.A., Liu, L.S.: Peridynamics simulations of geomaterial fragmentation by impulse loads. Int. J. Numer. Anal. Methods Geomech. 39, 1304–1330 (2015)

    Google Scholar 

  36. Sarego, G., Le, Q.V., Bobaru, F., Zaccariotto, M., Galvanetto, U.: Linearized state-based peridynamics for 2-D problems. Int. J. Numer. Methods Eng. 108(10), 1174–1197 (2016)

    MathSciNet  Google Scholar 

  37. Huang, D., Lu, G.D., Qiao, P.Z.: An improved peridynamic approach for quasi-static elastic deformation and brittle fracture analysis. Int. J. Mech. Sci. 94–95, 111–122 (2015)

    Google Scholar 

  38. Hu, W., Ha, Y.D., Bobaru, F.: Peridynamic model for dynamic fracture in unidirectional fiber-reinforced composites. Comput. Methods Appl. Mech. Eng. 217–220, 247–261 (2012)

    MathSciNet  MATH  Google Scholar 

  39. Xu, J.F., Askari, A., Weckner, O., Silling, S.A.: Peridynamic analysis of impact damage in composite laminates. J. Aerosp. Eng. 21, 187–194 (2008)

    Google Scholar 

  40. Azdoud, Y., Han, F., Lubineau, G.: A morphing framework to couple non-local and local anisotropic continua. Int. J. Solids Struct. 50, 1332–1341 (2013)

    Google Scholar 

  41. Azdoud, Y., Han, F., Lubineau, G.: The morphing method as a flexible tool for adaptive local/nonlocal simulation of static fracture. Comput. Mech. 54, 711–722 (2014)

    MathSciNet  MATH  Google Scholar 

  42. Galvanetto, U., Mudric, T., Shojaei, A., Zaccariotto, M.: An effective way to couple FEM meshes and peridynamics grids for the solution of static equilibrium problems. Mech. Res. Commun. 76, 41–47 (2016)

    Google Scholar 

  43. Kilic, B., Madenci, E.: Coupling of peridynamic theory and finite element method. J. Mech. Mater. Struct. 5, 707–733 (2010)

    Google Scholar 

  44. Liu, W.Y., Hong, J.W.: A coupling approach of discretized peridynamics with finite element method. Comput. Methods Appl. Mech. Eng. 245–246, 163–175 (2012)

    MathSciNet  MATH  Google Scholar 

  45. Zaccariotto, M., Mudric, T., Tomasi, D., Shojaei, A., Galvanetto, U.: Coupling of FEM meshes with peridynamic grids. Comput. Methods Appl. Mech. Eng. 330, 471–497 (2018)

    MathSciNet  Google Scholar 

  46. Li, H., Zhang, H.W., Zheng, Y.G., Ye, H.F., Lu, M.K.: An implicit coupling finite element and peridynanic method for dynamic problems of solids mechanics with crack propagation. Int. J. Appl. Mech. 10(4), 1850037 (2018)

    Google Scholar 

  47. Hou, T.Y., Wu, X.H.: A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134, 169–189 (1997)

    MathSciNet  MATH  Google Scholar 

  48. Hou, T.Y., Wu, X.H., Cai, Z.Q.: Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients. Math. Comput. 68, 913–943 (1999)

    MathSciNet  MATH  Google Scholar 

  49. Casadei, F., Rimoli, J.J., Ruzzene, M.: A geometric multiscale finite element method for the dynamic analysis of heterogeneous solids. Comput. Methods Appl. Mech. Engg. 263, 56–70 (2013)

    MATH  Google Scholar 

  50. Zhang, H.W., Fu, Z.D., Wu, J.K.: Coupling multiscale finite element method for consolidation analysis of heterogeneous saturated porous media. Adv. Water Resour. 32, 268–279 (2009)

    Google Scholar 

  51. Zhang, H.W., Wu, J.K., Fu, Z.D.: Extended multiscale finite element method for mechanical analysis of heterogeneous material. Acta Mech. Sin. 26, 899–920 (2010)

    MathSciNet  MATH  Google Scholar 

  52. Zhang, H.W., Wu, J.K., Fu, Z.D.: Extended multiscale finite element method for elasto-plastic analysis of 2D periodic lattice truss materials. Comput. Mech. 45, 623–635 (2010)

    MATH  Google Scholar 

  53. Li, H., Zhang, H.W., Zheng, Y.G.: A coupling extended multiscale finite element method for dynamic analysis of heterogeneous saturated porous media. Int. J. Numer. Methods Eng. 104, 18–47 (2015)

    MathSciNet  MATH  Google Scholar 

  54. Zhang, H.W., Lu, M.K., Zheng, Y.G., Zhang, S.: General coupling extended multiscale FEM for elasto-plastic consolidation analysis of heterogeneous saturated porous media. Int. J. Numer. Anal. Methods Geomech. 39, 63–95 (2015)

    Google Scholar 

  55. Zhang, S., Yang, D.S., Zhang, H.W., Zheng, Y.G.: Coupling extended multiscale finite element method for thermoelastic analysis of heterogeneous multiphase materials. Comput. Struct. 121, 32–49 (2013)

    Google Scholar 

  56. Leon, S.E., Paulino, G.H., Pereira, A., Menezes, I.F.M., Lages, E.N.: A unified library of nonlinear solution schemes. Appl. Mech. Rev. 64(4), 040803 (2012)

    Google Scholar 

  57. Trunk, B.: Einfluss der Bauteilgröße auf die Bruchenergie von Beton. Aedificatio Publishers, Freiburg (2000)

    Google Scholar 

  58. Su, X.T., Yang, Z.J., Liu, G.H.: Finite element modelling of complex 3D static and dynamic crack propagation by embedding cohesive elements in ABAQUS. Acta Mech. Solida Sin. 23(3), 271–282 (2010)

    Google Scholar 

  59. Madenci, E., Oterkus, E.: Peridynamic Theory and Its Applications. Springer, New York (2014)

    MATH  Google Scholar 

  60. Zaccariotto, M., Luongo, F., Sarego, G., Galvanetto, U.: Examples of applications of the peridynamic theory to the solution of static equilibrium problems. Aeronaut. J. 119, 677–700 (2015)

    Google Scholar 

  61. Carpinteri, A., Colombo, G.: Numerical analysis of catastrophic softening behavior (snap-back instability). Comput. Struct. 31(4), 607–636 (1989)

    Google Scholar 

  62. Bittencourt, T.N., Wawrzynek, P.A., Ingraffea, A.R., Sousa, J.L.: Quasi-automatic simulation of crack propagation for 2D LEFM problems. Eng. Fract. Mech. 55, 321–334 (1996)

    Google Scholar 

Download references

Acknowledgements

The supports from the National Natural Science Foundation of China (Nos. 11672062, 11772082 and 11672063), the LiaoNing Revitalization Talents Program (XLYC1807193), the 111 Project (No. B08014) and Fundamental Research Funds for the Central Universities are gratefully acknowledged.

Author information

Authors and Affiliations

  1. International Research Center for Computational Mechanics, State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Faculty of Vehicle Engineering and Mechanics, Dalian University of Technology, Dalian, 116024, People’s Republic of China

    Hongwu Zhang, Hui Li, Hongfei Ye & Yonggang Zheng

  2. Science and Technology on Reactor System Design Technology Laboratory, Nuclear Power Institute of China, No. 328 Changshun Avenue, Chengdu, 610200, Sichuan, People’s Republic of China

    Hui Li & Yixiong Zhang

Authors
  1. Hongwu Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hui Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hongfei Ye
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yonggang Zheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Yixiong Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Hongwu Zhang or Yonggang Zheng.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, H., Li, H., Ye, H. et al. A coupling extended multiscale finite element and peridynamic method for modeling of crack propagation in solids. Acta Mech 230, 3667–3692 (2019). https://doi.org/10.1007/s00707-019-02471-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-019-02471-2

Search

Navigation